A000391 -id:A000391 - cp's OEIS frontend

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583, 13602, 27613, 55579, 110445, 217554, 424148, 820294, 1572647, 2992892, 5652954, 10605608, 19765082, 36609945, 67405569, 123412204, 224728451, 407119735, 733878402, 1316631730, 2351322765, 4180714647, 7401898452, 13051476707, 22922301583, 40105025130, 69909106888, 121427077241, 210179991927, 362583131144

Offset: 0

N. J. A. Sloane

Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005
Euler transform of the triangular numbers 1,3,6,10,...
Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009
The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]
Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
Steven Finch, Integer Partitions, September 22, 2004, page 2. [Cached copy, with permission of the author]
Vaclav Kotesovec, Graph - The asymptotic ratio
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000293, A007294, A007326, A255939, A028377.
Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
Cf. also A278403 (log of o.g.f.).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> n*(n+1)/2): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2008

a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - Vladeta Jovovic, Sep 17 2002
a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 11 2015
G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 21 2018

More terms from Sascha Kurz, Aug 15 2002

A000335 Euler transform of A000292.

Original entry on oeis.org

1, 5, 15, 45, 120, 331, 855, 2214, 5545, 13741, 33362, 80091, 189339, 442799, 1023192, 2340904, 5302061, 11902618, 26488454, 58479965, 128120214, 278680698, 602009786, 1292027222, 2755684669, 5842618668, 12317175320, 25825429276, 53865355154, 111786084504, 230867856903, 474585792077, 971209629993

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..500
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p. 20.
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000391, A000417, A000428, A255965.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+2,3)): seq(a(n), n=1..26); # Alois P. Heinz, Sep 08 2008

max = 33; f[x_] := Exp[ Sum[ x^k/(1-x^k)^4/k, {k, 1, max}]]; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x], 1](* Jean-François Alcover, Nov 21 2011, after Joerg Arndt *)
nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]] (* Vaclav Kotesovec, Mar 11 2015 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, #*p[#] &]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Binomial[#+2, 3]&]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)

a(n)=if(n<1, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^4/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

N=66; x='x+O('x^66); gf=-1 + exp(sum(k=1, N, x^k/(1-x^k)^4/k)); Vec(gf) /* Joerg Arndt, Jul 06 2011 */

# uses[EulerTransform from A166861] and prepends a(0) = 1.
a = EulerTransform(lambda n: n*(n+1)*(n+2)//6)
print([a(n) for n in range(33)]) # Peter Luschny, Nov 17 2022

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 - Zeta(3) / (8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8*Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2 / (90*Zeta(5)) + Zeta'(-3)/6 + (Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5)) - Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5))) * n^(2/5) + Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Mar 12 2015

A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

Original entry on oeis.org

0, 0, 5, 3, 7, 8, 5, 7, 6, 3, 5, 7, 7, 7, 4, 3, 0, 1, 1, 4, 4, 4, 1, 6, 9, 7, 4, 2, 1, 0, 4, 1, 3, 8, 4, 2, 8, 9, 5, 6, 6, 4, 4, 3, 9, 7, 4, 2, 2, 9, 5, 5, 0, 7, 0, 5, 9, 4, 4, 7, 0, 2, 3, 2, 2, 3, 3, 2, 4, 5, 0, 1, 9, 9, 7, 9, 2, 4, 0, 6, 9, 5, 8, 6, 0, 9, 5, 1, 0, 3, 8, 7, 0, 8, 2, 5, 6, 8, 3, 2, 6, 7, 1, 2, 2, 4, 3

Offset: 0

Jean-François Alcover, Jun 18 2015

			0.0053785763577743011444169742104138428956644397422955070594470232233245...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15.1 Generalized Glaisher constants, p. 136-137.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A000335, A000391, A000417, A000428, A023872, A057527, A057528, A255050, A255052, A258350, A258351, A258352, A260404.

Join[{0, 0}, RealDigits[Zeta'[-3], 10, 105] // First]

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.
zeta'(-3) = -11/720 - log(A(3)), where A(3) is A243263.
Equals -11/720 + (gamma + log(2*Pi))/120 - 3*Zeta'(4)/(4*Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015

A000417 Euler transform of A000389.

Original entry on oeis.org

1, 7, 28, 105, 357, 1232, 4067, 13301, 42357, 132845, 409262, 1243767, 3727360, 11036649, 32300795, 93538278, 268164868, 761656685, 2144259516, 5986658951, 16583102077, 45593269265, 124464561544, 337479729179, 909156910290, 2434121462871, 6478440788169

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000335, A000391, A000428, A255965.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+4,5)): seq(a(n), n=1..30); # Alois P. Heinz, Sep 08 2008

nn = 100; b = Table[Binomial[n, 5], {n, 5, nn + 5}]; Rest[CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x]] (* T. D. Noe, Jun 20 2012 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^6/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

a(n) ~ (3*Zeta(7))^(31103/423360) / (2^(180577/423360) * sqrt(7*Pi) * n^(242783/423360)) * exp(Zeta'(-1)/5 - 5*Zeta(3)/(48*Pi^2) + Zeta(5)/(16*Pi^4) - Pi^36/(1162964338810860915 * Zeta(7)^5) + Pi^24 * Zeta(5) / (413420708484 * Zeta(7)^4) - Pi^22 / (137806902828 * Zeta(7)^3) - Pi^12 * Zeta(5)^2 / (551124 * Zeta(7)^3) + Pi^12 * Zeta(3) / (11252115 * Zeta(7)^2) + Pi^10 * Zeta(5) / (122472 * Zeta(7)^2) + 49*Zeta(5)^3 / (216 * Zeta(7)^2) - Pi^8 / (108864 * Zeta(7)) - Zeta(3) * Zeta(5) / (15*Zeta(7)) + Zeta'(-5)/120 + 7*Zeta'(-3)/24 + (22 * 2^(6/7) * Pi^30 / (46901442470561469 * 3^(1/7) * Zeta(7)^(29/7)) - 10 * 2^(6/7) * Pi^18 * Zeta(5) / (8931928887 * 3^(1/7) * Zeta(7)^(22/7)) + Pi^16 / (141776649 * 6^(1/7) * Zeta(7)^(15/7)) + 2^(6/7) * Pi^6 * Zeta(5)^2 / (1701 * 3^(1/7) * Zeta(7)^(15/7)) - 2^(6/7) * Pi^6 * Zeta(3) / (19845 * 3^(1/7) * Zeta(7)^(8/7)) - Pi^4 * Zeta(5) / (216 * 6^(1/7) * Zeta(7)^(8/7))) * n^(1/7) + (-2 * 2^(5/7) * Pi^24 / (3938980639167 * 3^(2/7) * Zeta(7)^(23/7)) + Pi^12 * Zeta(5) / (500094 * 6^(2/7) * Zeta(7)^(16/7)) - Pi^10 / (142884 * 6^(2/7) * Zeta(7)^(9/7)) - 7*Zeta(5)^2 / (12 * 6^(2/7) * Zeta(7)^(9/7)) + Zeta(3)/(5 * (6*Zeta(7))^(2/7))) * n^(2/7) + (5 * 2^(4/7) * Pi^18 / (8931928887 * 3^(3/7) * Zeta(7)^(17/7)) - Pi^6 * Zeta(5) / (567 * 6^(3/7) * Zeta(7)^(10/7)) + Pi^4 / (108 * (6*Zeta(7))^(3/7))) * n^(3/7) + (-Pi^12 / (750141 * 6^(4/7) * Zeta(7)^(11/7)) + 7*Zeta(5) / (4 * (6 * Zeta(7))^(4/7))) * n^(4/7) + 2^(2/7) * Pi^6 / (945 * (3*Zeta(7))^(5/7)) * n^(5/7) + 7*Zeta(7)^(1/7) / 6^(6/7) * n^(6/7)). - Vaclav Kotesovec, Mar 12 2015

More terms from Sean A. Irvine, Nov 14 2010

A000428 Euler transform of A000579.

Original entry on oeis.org

1, 8, 36, 148, 554, 2094, 7624, 27428, 96231, 332159, 1126792, 3769418, 12437966, 40544836, 130643734, 416494314, 1314512589, 4110009734, 12737116845, 39144344587, 119350793207, 361173596536, 1085171968872

Offset: 1

N. J. A. Sloane

nonn

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^binomial(k+m-2,m-1) and m >= 1, then log(a(n)) ~ (m+1) * Zeta(m+1)^(1/(m+1)) * (n/m)^(m/(m+1)). - Vaclav Kotesovec, Mar 12 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Vaclav Kotesovec, Asymptotic formula
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000335, A000391, A000417, A255965.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+5,6)): seq(a(n), n=1..30); # Alois P. Heinz, Sep 08 2008

nn = 30; b = Table[Binomial[n, 6], {n, 6, nn + 6}]; Rest[CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x]] (* T. D. Noe, Jun 20 2012 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^7/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

A255965 Expansion of Product_{k>=1} 1/(1-x^k)^binomial(k+6,7).

Original entry on oeis.org

1, 1, 9, 45, 201, 819, 3357, 13329, 52215, 199686, 750733, 2774793, 10112184, 36357280, 129131448, 453379226, 1574884565, 5415956550, 18450934294, 62303210591, 208624947952, 693066815809, 2285129922950, 7480504628754, 24320897894515, 78557786077315

Offset: 0

Vaclav Kotesovec, Mar 12 2015

nonn

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^binomial(k+m-2,m-1) and m >= 1, then log(a(n)) ~ (m+1) * Zeta(m+1)^(1/(m+1)) * (n/m)^(m/(m+1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic formula

Cf. A000041 (m=1), A000219 (m=2), A000294 (m=3), A000335 (m=4), A000391 (m=5), A000417 (m=6), A000428 (m=7).

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)/7!),{k,1,nmax}],{x,0,nmax}],x]

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)^8)). - Ilya Gutkovskiy, May 28 2018

A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 7, 27, 98, 323, 1085, 3471, 10998, 33874, 102737, 305849, 897899, 2597822, 7423408, 20957775, 58524868, 161741013, 442705279, 1200718351, 3228796864, 8611973548, 22793714865, 59887897679, 156252738062, 404964879419, 1042884107691, 2669317020743, 6792321636929

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

Euler transform of A002415, shifted left one place.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A000391, A002415, A023871, A279215, A290792.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d^2*
      (d+2)*(d+1)^2/12, d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 07 2018

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((k + 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2 (d + 2)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002415(k+1).
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^5)).
a(n) ~ exp(Zeta'(-1)/6 - Zeta(3) / (4*Pi^2) + 149*Zeta(5) / (32*Pi^4) + 15876 * Zeta(3) * Zeta(5)^2 / Pi^12 - 666792 * Zeta(5)^3 / Pi^14 + 108884466432 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/3 + (-7*(7/2)^(1/6) * Pi / (384*sqrt(3)) - 21 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / Pi^7 + 3087 * sqrt(3) * (7/2)^(1/6) * Zeta(5)^2 / (2*Pi^9) - 30339036 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(5)^4 / Pi^19) * n^(1/6) + ((7/2)^(1/3) * Zeta(3) / (2*Pi^2) - 21 * (7/2)^(1/3) * Zeta(5) / (2*Pi^4) + 254016 * 2^(2/3) * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (sqrt(7/6) * Pi / 12 - 756 * sqrt(42) * Zeta(5)^2 / Pi^9) * sqrt(n) + (9 * 2^(1/3) * 7^(2/3) * Zeta(5) / Pi^4) * n^(2/3) + (2 * (2/7)^(1/6) * sqrt(3) * Pi) / 5 * n^(5/6)) * Pi^(1/90) / (2^(247/270) * 3^(34/45) * 7^(23/270) * n^(79/135)). - Vaclav Kotesovec, Jun 08 2018

A007328 Difference between the number of 5-dimensional partitions of n and an approximation derived from binomial(n,4).

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 75, 310, 1060, 3281, 9564, 26719, 72239, 191569, 500797, 1299925, 3362473, 8697198, 22513878, 58352126, 151267141, 391728632, 1011734975, 2602330120, 6657204192, 16920629023, 42697311397, 106912113623, 265560809521, 654270114555

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100; alternative link.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Cf. A000390, A000391.

Cf. A007326, A007327, A007329, A007330.

a(n) = A000391(n) - A000390(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(21) from Sean A. Irvine, Dec 18 2017

More terms from Amiram Eldar, May 11 2024

A264925 G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)(n+2)(n+3)*(n+4)/4!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 215, 360, 605, 1080, 2003, 3890, 7570, 14715, 27960, 52255, 95705, 173295, 311060, 557400, 999032, 1795880, 3235130, 5835955, 10521060, 18931287, 33956485, 60692510, 108087835, 191883595, 339724144, 600203700, 1058605775, 1864535670, 3279862975, 5762287759, 10109925380

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 5 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^5 + 5*x^6 + 15*x^7 + 35*x^8 + 70*x^9 + 127*x^10 + 215*x^11 + 360*x^12 +...
where
1/A(x) = (1-x^5) * (1-x^6)^5 * (1-x^7)^15 * (1-x^8)^35 * (1-x^9)^70 * (1-x^10)^126 * (1-x^11)^210 * (1-x^12)^330 * (1-x^13)^495 *...
Also,
log(A(x)) = (x/(1-x))^5 + (x^2/(1-x^2))^5/2 + (x^3/(1-x^3))^5/3 + (x^4/(1-x^4))^5/4 + (x^5/(1-x^5))^5/5 + (x^6/(1-x^6))^5/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A052847, A264923, A264924, A264926.

Cf. A000391, A255052.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-4)*(k-3)*(k-2)*(k-1)/24), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)*(d-4)/4!)}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^5 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)*(d-4)/4!.
a(n) ~ Pi^(95/288) / (2 * 3^(527/576) * 7^(239/1728) * n^(1103/1728)) * exp(-25*Zeta'(-1)/12 - log(2*Pi)/2 + 595*Zeta(3)/(48*Pi^2) - 29291*Zeta(5) / (128*Pi^4) - 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) + 72930375 * Zeta(5)^3 / (2*Pi^14) - 1063324867500 * Zeta(5)^5/Pi^24 - 5*Zeta'(-3)/12 + (41 * 7^(1/6) * Pi/(768*sqrt(3)) - 2625 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5)/(2*Pi^7) + 540225 * sqrt(3) * 7^(1/6) * Zeta(5)^2/(16*Pi^9) - 4740474375 * sqrt(3) * 7^(1/6) * Zeta(5)^4/(4*Pi^19)) * n^(1/6) + (-25 * 7^(1/3) * Zeta(3)/(4*Pi^2) + 735 * 7^(1/3) * Zeta(5) /(8*Pi^4) - 3969000 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (7*sqrt(7/3)*Pi/24 - 4725 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) - 45 * 7^(2/3) * Zeta(5)/(2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - Vaclav Kotesovec, Dec 09 2015

cp's OEIS Frontend

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A000335 Euler transform of A000292.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A000417 Euler transform of A000389.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A000428 Euler transform of A000579.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A255965 Expansion of Product_{k>=1} 1/(1-x^k)^binomial(k+6,7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

Views

Author

Keywords

A264925 G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)(n+2)(n+3)*(n+4)/4!).